The Effect of Trimming on the Strong Law of Large Numbers

‘Trimmed’ sample sums may be defined for r = 1, 2, …, byS n( r )= S n − M n( 1 )− M n( 2 )− … − M n( r )andS n( r )= S n − X n( 1 )− X n( 2 )− … − X n( r )where S n = X 1 + X 2 + … + X n is the sum of independent and identically distributed random variables X i ,M n ( 1 )⩾ …   ⩾ M n ( n )denote X l , …, X n arranged in decreasing order, andX n ( j )is the observation with the j th largest modulus. We investigate the effects of these kinds of trimming on various forms of convergence and divergence of the sample sum. In particular, we provide integral tests for ( r ) S n / n → ±∞, and analytical criteria for almost sure relative stability when the number of points trimmed, r , is fixed, but n → ∞. Some surprising results occur. For example, when r = 0, 1, 2, …, ( r ) S n may be almost surely negatively relatively stable ( ( r ) S n / B n → −1 a.s. as n → ∞ for some non‐stochastic sequence B n ↑ ∞) only if −∞ < EX 1 ⩽ 0, and a striking corollary of this is an example of a random walk S n which is recurrent (even has mean 0), but for which ( r ) S n andS ˜ nare transient when r ⩾ 1.